Assignment 8

Due: 10:00 a.m., Friday, Nov. 13

1. The atomic orbital is obtained by taking a linear combination of the and orbitals. The result is

   

   As written, the wavefunction is not normalized.

   1. Normalize this wavefunction.
   2. Calculate the average kinetic energy.
   3. Plot the radial probability density.
   4. Calculate the probability that for a hydrogen atom in this state. Give a floating-point answer with at least four significant figures.

   Hints: Maple will need to know that . The volume element in spherical polar coordinates is

   

   The angles and run over the following ranges: , . The Laplacian operator is spherical polar coordinates is

   

   To plot the radial probability density, use the following Maple command: plot(subs(a0=1,rpd),r=0..?); where rpd is your radial probability density and ? is replaced by some suitable number.

2. Suppose that we have a chemical reaction for which the potential energy along the reaction coordinate (x) has the following form:

   

   1. Plot the potential energy. Note the symmetric double well shape with an intervening energy barrier. This might for instance be the potential energy profile for a proton transfer between two chemically similar group. Such reactions are common in the active sites of enzymes.

      Maple hint: To scale the x axis in units of and the energy axis in units of , use the following plot command: plot(subs(xe=1,Ea=1,V(x)),x=-?..?); where ? is replaced by a suitable number.

   2. Write down the Hamiltonian for a proton subjected to this potential energy.
   3. The probability density should have maxima corresponding to the two potential energy wells. A reasonable guess for the wavefunction would be a superposition of Gaussian functions with their maxima at . Therefore use the following variational wavefunction to find an approximation to the ground state wavefunction:

      

      Maple hints: b>0. Maple's answer will be a complicated expression involving the RootOf() operator. Save this expression in a variable, i.e. type something like best_b := solve(...);.

   4. Suppose that we are dealing with a proton transfer reaction for which and . The mass of a proton is . Define these values in your Maple worksheet, along with , and use evalf() to find the best value of b. Don't assign this value to b. Instead, use subs() to substitute it into your variational energy, and evalf() to get a floating-point value. Store this value in a variable with a distinctive name (say, E0).
   5. Find the points where , where is your approximation to the ground state energy. Then calculate the probability that the proton is to be found in the activation barrier between the two wells. In your opinion, is tunneling an important process for this system?

      Maple hint: You will have to use subs() to substitute b into your wavefunction.